Average Error: 33.3 → 9.7
Time: 16.5s
Precision: 64
\[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b_2 \le -6.778998599359054 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 2.1478963185043958 \cdot 10^{-57}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]
\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}
↓
\begin{array}{l}
\mathbf{if}\;b_2 \le -6.778998599359054 \cdot 10^{+108}:\\
\;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\

\mathbf{elif}\;b_2 \le 2.1478963185043958 \cdot 10^{-57}:\\
\;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\

\end{array}
double f(double a, double b_2, double c) {
        double r332446 = b_2;
        double r332447 = -r332446;
        double r332448 = r332446 * r332446;
        double r332449 = a;
        double r332450 = c;
        double r332451 = r332449 * r332450;
        double r332452 = r332448 - r332451;
        double r332453 = sqrt(r332452);
        double r332454 = r332447 + r332453;
        double r332455 = r332454 / r332449;
        return r332455;
}
↓
double f(double a, double b_2, double c) {
        double r332456 = b_2;
        double r332457 = -6.778998599359054e+108;
        bool r332458 = r332456 <= r332457;
        double r332459 = -2.0;
        double r332460 = a;
        double r332461 = r332456 / r332460;
        double r332462 = 0.5;
        double r332463 = c;
        double r332464 = r332463 / r332456;
        double r332465 = r332462 * r332464;
        double r332466 = fma(r332459, r332461, r332465);
        double r332467 = 2.1478963185043958e-57;
        bool r332468 = r332456 <= r332467;
        double r332469 = r332456 * r332456;
        double r332470 = r332463 * r332460;
        double r332471 = r332469 - r332470;
        double r332472 = sqrt(r332471);
        double r332473 = r332472 - r332456;
        double r332474 = 1.0;
        double r332475 = r332474 / r332460;
        double r332476 = r332473 * r332475;
        double r332477 = -0.5;
        double r332478 = r332477 * r332464;
        double r332479 = r332468 ? r332476 : r332478;
        double r332480 = r332458 ? r332466 : r332479;
        return r332480;
}

Error

Bits error versus a

Bits error versus b_2

Bits error versus c

Derivation

  1. Split input into 3 regimes

  2. if b_2 < -6.778998599359054e+108

    1. Initial program 46.7

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified46.7

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around -inf 3.3

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b_2} - 2 \cdot \frac{b_2}{a}}\]

    4. Simplified3.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)}\]

    if -6.778998599359054e+108 < b_2 < 2.1478963185043958e-57

    1. Initial program 13.1

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified13.1

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]
    3. Using strategy rm

    4. Applied div-inv13.2

      \[\leadsto \color{blue}{\left(\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2\right) \cdot \frac{1}{a}}\]

    if 2.1478963185043958e-57 < b_2

    1. Initial program 52.9

      \[\frac{\left(-b_2\right) + \sqrt{b_2 \cdot b_2 - a \cdot c}}{a}\]

    2. Simplified52.9

      \[\leadsto \color{blue}{\frac{\sqrt{b_2 \cdot b_2 - a \cdot c} - b_2}{a}}\]

    3. Taylor expanded around inf 8.0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b_2}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;b_2 \le -6.778998599359054 \cdot 10^{+108}:\\ \;\;\;\;\mathsf{fma}\left(-2, \frac{b_2}{a}, \frac{1}{2} \cdot \frac{c}{b_2}\right)\\ \mathbf{elif}\;b_2 \le 2.1478963185043958 \cdot 10^{-57}:\\ \;\;\;\;\left(\sqrt{b_2 \cdot b_2 - c \cdot a} - b_2\right) \cdot \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{2} \cdot \frac{c}{b_2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 +o rules:numerics
(FPCore (a b_2 c)
  :name "quad2p (problem 3.2.1, positive)"
  (/ (+ (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))